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The Banach Space C0 E extracta mathematicae Vol. 16, N´um.1, 1 – 25 (2001) The Banach Space c0 Gilles Godefroy Equipe d’Analyse, Universit´eParis VI, Case 186, 4, Place Jussien, 75252 Paris Cedex 05, France e-mail: [email protected] (Survey paper presented by J.P. Moreno) AMS Subject Class. (2000): 46B20 Received October 22, 2000 I. Introduction We survey in these notes some recent progress on the understanding of the Banach space c0 an of its subspaces. We did not try to complete the (quite ambitious) task of writing a comprehensive survey. A few topics have been selected, in order to display the variety of techniques which are required in such investigations. It is hoped that this selection can provide the reader with some intuition about the behaviour of the space. Our choice has also been influenced by the open problems which conclude these notes. It is our hope that some of these problems are not desperately hard. The unit ball of c0 can be (vaguely) visualized as a cubic box with no vertices; somehow, only the internal parts of the sides remain. Cubic boxes are suitable for packing, and indeed it is easy to map a space to c0 (see x II). A cube is a polyhedron, and indeed the polyhedral nature of c0 and its subspaces inform us of some of their isometric properties (see x III). The space c0 contains “very few” compact sets, unlike spaces which do not contain c0 and retain some features of reflexive spaces (see x IV). The natural norm of c0 is flat at every point in the direction of a finite-codimensional space and c0 is the largest space with this property. This helps to prove that the unit ball of c0 is a sturdy box, whose shape cannot be altered by non-linear Lipschitz maps (see x V). Finally, vectors in c0 can easily be approximated by restricting their support to given finite subsets. Subspaces of c0 are in general not stable under this operation. However, smoothness properties of these subspaces sometimes help to build a decent approximation scheme (see x VI). A sample of open problems is displayed in the last section VII. 1 2 g. godefroy Notation We denote by c0 the separable Banach space of all real se- quences (un) such that limn!1(un) = 0. This space is equipped with its canonical norm © ª k(un)k1 = sup junj; n ¸ 1 : Capital letters such as X, Y , ::: will denote Banach spaces. By “sub- ¤ space”, we mean “closed subspace”. We denote by en the linear form defined on c0 by ¤ en((uk)) = un: ¤ ¤ 1 Of course, en 2 c0 = ` . We denote by NA(X) the subset of the dual space X¤ consisting of norm-attaining linear functionals. We otherwise follow the classical notation as it can be found e.g. in [32]. II. c0 is an easy target It is quite easy to give a usable representation for operators T from a Banach space X to c0. Indeed, if we let ¤ ¤ ¤ ¤ T (en) = xn 2 X then clearly ¤ T (x) = (xn(x))n¸1 (1) ¤ ¤ ¤ ¤ and the sequence (xn) is w -convergent to 0 in X . Conversely, if (xn) is a ¤ ¤ w -null sequence in X , then (1) defines an operator from X to c0. This very simple representation provides us with a bunch of operators from X into c0, at least when the space X is separable since in this case, bounded subsets of X¤ are w¤-metrizable and thus convergent sequences are easily constructed. This simple idea lies in the background of an old but important result on c0: Sobczyk’s theorem. Theorem II.1. ([35]) Let X be a separable Banach space, and Y a sub- space of X. Let T : Y ! c0 be a bounded operator. There exists Te : X ! c0 such that e (i) TjY = T . (ii) kTek · 2kT k. the banach space c0 3 ¤ ¤ ¤ ¤ Proof. ([36]) Let yn = T (en). Clearly kT k = supn kynk. By Hahn– ¤ ¤ ¤ ¤ ¤ Banach theorem, there exist xn 2 X such that kxnk · kT k and xnjY = yn for all n ¸ 1. We set K = fx¤ 2 X¤; kx¤k · kT kg and L = K \ Y ?. When equipped with the w¤-topology, K is metrizable compact. Let d be a distance which defines the weak-star topology on K. ¤ ¤ ¤ Since lim xn(y) = lim yn(y) = 0 for every y 2 Y , every w -cluster point to ¤ the sequence (xn) belongs to L. It follows by compactness that the distance ¤ ¤ from (xn) to L in (K; d) tends to 0, hence we can pick tn 2 L such that ¤ ¤ ¤ w - lim(xn ¡ tn) = 0: If we now define Te : X ! c0 by ¡ ¢ e ¤ ¤ T (x) = (xn ¡ tn)(x) n¸1 it is easily seen that Te works. Moreover ¡ ¢ e ¤ ¤ ¤ ¤ kT k = sup kxn ¡ tnk · sup kxnk + ktnk · 2kT k: n n Let us spell out two important applications. Corollary II.2. Let X be a separable Banach space, and let Y be a subspace which is isomorphic to c0. Then Y is complemented in X. Proof. Let T : Y ! c0 be an isomorphism onto c0. In the notation of Theorem II.1, T ¡1 ± Te : X ! Y is a projection. Corollary II.3. Let X be a separable Banach space which has a sub- space Y such that both spaces Y and X=Y are isomorphic to subspaces of c0. Then X is isomorphic to a subspace of c0. Proof. Let T : Y ! c0 and S : X=Y ! c0 be isomorphisms onto sub- spaces of c0. We denote again Te : X ! c0 an operator extending T , and Q : X ! X=Y the canonical¡ map. It is easily¢ checked that V : X ! c0 © c0 ' c0 defined by V (x) = Te(x);S ± Q(x) is an isomorphism of X into c0. 4 g. godefroy Remark II.4. Sobczyk’s theorem, and the above corollaries, fail in the nonseparable case. Indeed we can take X = `1 in corollary II.2 and c0 is not complemented in `1. There is a compact set K [9, Example VI.8.7.], such that C(K) = X contains a subspace Y isomorphic to c0(N) such that X=Y is isomorphic to c0(Γ), with jΓj = c, but X is not a subspace of c0(Γ) since X is not weakly compactly generated. Although it is harder to map a nonseparable Banach space into a c0(Γ) space, such mappings play a fundamental role in the study of “nice” nonsep- arable spaces. We refer to the last three chapters of [9] for this topic. III. Some isometric properties It is easy to characterize spaces which are isometric to subspaces of c0 by applying the formula (1) from x II to an isometric embedding. Proposition III.1. Let X be a Banach space. The following assertions are equivalent: (i) X is isometric to a subspace of c0. ¤ ¤ ¤ ¤ (ii) There exists a sequence of (xn) in X such that w -limn(xn) = 0 and for all x 2 X, © ¤ ª kxk = sup jxn(x)j; n ¸ 1 : The proof follows immediately from the consideration of ¡ ¤ ¢ T (x) = xn(x) n¸1 ¤ which is an isometric embedding into c0 if and only if (xn) satisfies (ii). This simple proposition has some interesting consequences. We first need a defini- tion. Definition III.2. Let X be a Banach space, and g : X ! R a continuous function. We say that g locally depends upon finitely many coordinates if for ¤ every x 2 X, there exist " > 0, a finite subset ff1; f2; : : : ; fng of X and a n continuous function ' : R ! R such that g(y) = '(f1(y); f2(y); : : : ; fn(y)) for every y such that kx ¡ yk < ". When g : X ! R is a norm, we say that it locally depends upon finitely many coordinates if the above condition holds for every x 2 X n f0g. We note the banach space c0 5 that when this happens, ' can be chosen to be a norm on Rn. Indeed, let T : X ! Rn be defined by ¡ ¢ T (y) = fi(y) 1·i·n and pick any y 2 SX with kx ¡ yk < ". For every u 2 Ker T , the function ¸ ¡! ky + ¸uk is convex, and ¸ 1 on some neighbourhood of 0, hence ¸ 1 on R and we have kykX= Ker T = 1: It follows that we can take '(¢) = k ¢ kX= Ker T . Proposition III.3. Let X be a subspace of c0, and let k ¢ k be the re- striction of k ¢ k1 to X. Then, for any x 2 X n f0g, there are ± > 0 and a ¤ ¤ finite subset (xn)n2J of X such that if kx ¡ yk < ±, then © ¤ ª kyk = sup jxn(y)j; n 2 J : In particular, k ¢ k depends locally of finitely many coordinates. Moreover, X is polyhedral; that is, the unit ball of every finite dimensional subspace of X has finitely many extreme points. Proof. Pick x 2 X n f0g. We have ¤ kxk = sup jen(x)j n ¤ and limn en(x) = 0. Hence if ¤ I = fn ¸ 1; jen(x)j < kxkg one has © ¤ ª kxk ¡ sup jen(x)j; n 2 I = " > 0 and the set J = N n I is finite. If y 2 X is such that kx ¡ yk < "=3, one clearly has © ¤ ª kyk = sup jen(y)j; y 2 J and this show our first assertion. Note that this assertion really means that the norm is locally linear when J is a singleton. 6 g. godefroy For showing polyhedrality, we pick E ⊆ X a finite dimensional subspace.
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